Conjugacy is Equivalence Relation

Theorem

Conjugacy of group elements is an equivalence relation.

Proof

Checking each of the criteria for an equivalence relation in turn:

Reflexive

$\forall x \in G: e_G \circ x = x \circ e_G \implies x \sim x$

Thus conjugacy of group elements is reflexive.

$\Box$

Symmetric

$\ds $

$\ds x \sim y$

$\ds $

$\leadsto$

$\ds a \circ x = y \circ a$

Definition of Conjugate of Group Element

$\ds $

$\leadsto$

$\ds a \circ x \circ a^{-1} = y$

Definition of Group

$\ds $

$\leadsto$

$\ds a^{-1} \circ y = x \circ a^{-1}$

Definition of Group

$\ds $

$\leadsto$

$\ds y \sim x$

Definition of Conjugate of Group Element

Thus conjugacy of group elements is symmetric.

$\Box$

Transitive

$\ds $

$\ds x \sim y, y \sim z$

$\ds $

$\leadsto$

$\ds a_1 \circ x = y \circ a_1, a_2 \circ y = z \circ a_2$

Definition of Conjugate of Group Element

$\ds $

$\leadsto$

$\ds a_2 \circ a_1 \circ x = a_2 \circ y \circ a_1 = z \circ a_2 \circ a_1$

Definition of Group

$\ds $

$\leadsto$

$\ds x \sim z$

Definition of Conjugate of Group Element

Thus conjugacy of group elements is transitive.

$\Box$

All criteria are satisfied, and so conjugacy of group elements is shown to be an equivalence relation.

$\blacksquare$

Sources

1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 10$. Equivalence Relations
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Exercise $25.16 \ \text{(a)}$
1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Exercise $5.16 \ \text{(i)}$
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 51$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 48.1$ Conjugacy
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conjugate elements
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): conjugate (conjugate elements)